J. Aracena, J. Demongeot, and E. Goles, Positive and Negative Circuits in Discrete Neural Networks, IEEE Transactions on Neural Networks, vol.15, issue.1, pp.77-83, 2004.
DOI : 10.1109/TNN.2003.821555

K. Saeed-akhoondian-amiri, S. Kawarabayashi, P. Kreutzer, and . Wollan, The Erd? os?Pósa property for directed graphs ArXiv preprint arXiv, pp.904-0727, 2016.

[. Lakshmanan, C. Bujtás, and Z. Tuza, Small Edge Sets Meeting all Triangles of a Graph, Graphs and Combinatorics, vol.26, issue.3, pp.381-392, 2011.
DOI : 10.1112/blms/26.4.321

A. Alon, Piercing d -Intervals, Discrete & Computational Geometry, vol.19, issue.3, pp.333-334, 1998.
DOI : 10.1007/PL00009349

A. Alon, Covering a hypergraph of subgraphs, Discrete Mathematics, vol.257, issue.2-3, pp.249-254, 2002.
DOI : 10.1016/S0012-365X(02)00427-2

J. Aracena, Maximum Number of Fixed Points in Regulatory Boolean Networks, Bulletin of Mathematical Biology, vol.19, issue.5, pp.1398-1409, 2008.
DOI : 10.1002/bbpc.19940980916

J. Aracena, A. Richard, and L. Salinas, Number of Fixed Points and Disjoint Cycles in Monotone Boolean Networks, SIAM Journal on Discrete Mathematics, vol.31, issue.3, 2016.
DOI : 10.1137/16M1060868
URL : https://hal.archives-ouvertes.fr/hal-01630477

E. Birmelé, J. A. Bondy, and B. A. Reed, The Erd??s???P??sa Property For Long Circuits, Combinatorica, vol.27, issue.2, pp.135-145, 2007.
DOI : 10.1007/s00493-007-0047-0

L. Hans, F. V. Bodlaender, D. Fomin, E. Lokshtanov, S. Penninkx et al., Meta) Kernelization. ArXiv preprint 0904.0727, full version of, 2009.

L. Hans, F. V. Bodlaender, D. Fomin, E. Lokshtanov, S. Penninkx et al., Proceedings of FOCS 2009, pp.629-638, 2009.

J. Battle, F. Harary, and Y. Kodama, Additivity of the genus of a graph, Bulletin of the American Mathematical Society, vol.68, issue.6, pp.565-568
DOI : 10.1090/S0002-9904-1962-10847-7

F. Henning-bruhn, O. Joos, and . Schaudt, Long cycles through prescribed vertices have the Erd? os-Pósa property. ArXiv preprint arXiv, pp.1412-2894, 2014.

[. Bousquet, Hitting sets: VC-dimension and Multicut, 2013.

C. Berge and B. Reed, Optimal packings of edge-disjoint odd cycles, Discrete Mathematics, vol.211, issue.1-3, pp.197-202, 2000.
DOI : 10.1016/S0012-365X(99)00283-6

N. Bousquet and S. Thomassé, VC-dimension and Erd??s???P??sa property, Discrete Mathematics, vol.338, issue.12, pp.2302-2317, 2015.
DOI : 10.1016/j.disc.2015.05.026

C. Chekuri and J. Chuzhoy, Large-treewidth graph decompositions and applications. ArXiv preprint arXiv:1304.1577, full version of, 2013.

C. Chekuri and J. Chuzhoy, Large-treewidth graph decompositions and applications, Proceedings of the 45th annual ACM symposium on Symposium on theory of computing, STOC '13, pp.291-300, 2013.
DOI : 10.1145/2488608.2488645

C. Chekuri and J. Chuzhoy, Polynomial Bounds for the Grid-Minor Theorem ArXiv preprint arXiv:1305.6577, full version of, 2013.

C. Chekuri and J. Chuzhoy, Polynomial bounds for the grid-minor theorem, Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pp.60-69, 2014.

[. Chudnovsky, A. Fradkin, and P. Seymour, Tournament immersion and cutwidth, Journal of Combinatorial Theory, Series B, vol.102, issue.1, pp.93-101, 2012.
DOI : 10.1016/j.jctb.2011.05.001

M. Chudnovsky, J. Geelen, B. Gerards, L. Goddyn, M. Lohman et al., Packing Non-Zero A-Paths In Group-Labelled Graphs, Combinatorica, vol.26, issue.5, pp.521-532, 2006.
DOI : 10.1007/s00493-006-0030-1

[. Chatzidimitriou, J. Raymond, I. Sau, and D. M. Thilikos, An O(log OP T )-approximation for covering and packing minor models of ? r . ArXiv preprint, 2015.

[. Chatzidimitriou, J. Raymond, I. Sau, and D. M. Thilikos, An O(log opt)-approximation for covering
URL : https://hal.archives-ouvertes.fr/lirmm-01609998

, Approximation and Online Algorithms: 13th International Workshop, pp.122-132, 2015.

M. Chudnovsky and P. Seymour, A well-quasi-order for tournaments, Journal of Combinatorial Theory, Series B, vol.101, issue.1, pp.47-53, 2011.
DOI : 10.1016/j.jctb.2010.10.003

R. Diestel, Graph Theory, volume 173 of Graduate Texts in Mathematics, 2005.

[. Diestel, K. Kawarabayashi, and P. Wollan, The Erd??s???P??sa property for clique minors in highly connected graphs, Journal of Combinatorial Theory, Series B, vol.102, issue.2, pp.454-469, 2012.
DOI : 10.1016/j.jctb.2011.08.001
URL : https://doi.org/10.1016/j.jctb.2011.08.001

[. Dejter and V. Neumann-lara, Unboundedness for generalized odd cyclic transversality, Colloq. Math. Soc. János Bolyai, pp.195-203, 1987.

G. Ding and B. Oporowski, On tree-partitions of graphs, Discrete Mathematics, vol.149, issue.1-3, pp.45-58, 1996.
DOI : 10.1016/0012-365X(94)00337-I
URL : https://doi.org/10.1016/0012-365x(94)00337-i

G. Ding, Z. Xu, and W. Zang, Packing cycles in graphs, II, Journal of Combinatorial Theory, Series B, vol.87, issue.2, pp.244-253, 2003.
DOI : 10.1016/S0095-8956(02)00007-2
URL : https://doi.org/10.1016/s0095-8956(02)00007-2

G. Ding and W. Zang, Packing Cycles in Graphs, Journal of Combinatorial Theory, Series B, vol.86, issue.2, pp.381-407, 2002.
DOI : 10.1006/jctb.2002.2134
URL : https://doi.org/10.1006/jctb.2002.2134

P. Erd?, O. , and L. Pósa, On independent circuits contained in a graph, Canadian Journal of Mathematics, vol.17, pp.347-352, 1965.

S. Fiorini and A. Herinckx, A Tighter Erd??s-P??sa Function for Long Cycles, Journal of Graph Theory, vol.41, issue.1, pp.111-116, 2014.
DOI : 10.1016/0095-8956(86)90030-4

S. Fiorini, N. Hardy, B. Reed, and A. Vetta, Approximate minmax relations for odd cycles in planar graphs, Integer Programming and Combinatorial Optimization, pp.35-50, 2005.
DOI : 10.1007/11496915_4
URL : http://cgm.cs.mcgill.ca/~reedbook/papers/ApproximateMinMaxRelations.pdf

[. Fiorini, G. Joret, and I. Sau, Optimal Erd? os?Pósa property for pumpkins, 2013.

[. Fiorini, G. Joret, and D. R. Wood, Excluded Forest Minors and the Erd??s???P??sa Property, Combinatorics, Probability and Computing, vol.22, issue.05, pp.700-721, 2013.
DOI : 10.1007/978-3-642-23719-5

V. Fedor, D. Fomin, N. Lokshtanov, G. Misra, S. Philip et al., Quadratic upper bounds on the erd? os?pósa property for a generalization of packing and covering cycles, Journal of Graph Theory, vol.74, issue.4, pp.417-424, 2013.

A. Fradkin and P. Seymour, Tournament pathwidth and topological containment, Journal of Combinatorial Theory, Series B, vol.103, issue.3, pp.374-384, 2013.
DOI : 10.1016/j.jctb.2013.03.001
URL : https://doi.org/10.1016/j.jctb.2013.03.001

V. Fedor, S. Fomin, D. M. Saurabh, and . Thilikos, Strengthening Erd? os? Pósa property for minor-closed graph classes, Journal of Graph Theory, vol.66, issue.3, pp.235-240, 2011.

T. Gallai, Maximum-minimum sätze und verallgemeinerte faktoren von graphen, Acta Mathematica Hungarica, vol.12, p.3, 1964.
DOI : 10.1007/bf02066678

J. Geelen and K. Kabell, The Erd??s???P??sa property for matroid circuits, Journal of Combinatorial Theory, Series B, vol.99, issue.2, pp.407-419, 2009.
DOI : 10.1016/j.jctb.2008.08.004
URL : https://doi.org/10.1016/j.jctb.2008.08.004

[. Giannopoulou, O. Kwon, J. Raymond, and D. M. Thilikos, Packing and covering immersion models of planar subcubic graphs. arXiv preprint arXiv:1602.04042, full version of, 2016.
DOI : 10.1007/978-3-662-53536-3_7
URL : https://hal.archives-ouvertes.fr/lirmm-01370310

[. Giannopoulou, O. Kwon, J. Raymond, and D. M. Thilikos, Packing and Covering Immersion Models of Planar Subcubic Graphs, Graph-Theoretic Concepts in Computer Science: 42nd International Workshop, WG 2016, p.74, 2016.
DOI : 10.1016/j.jctb.2014.07.003
URL : https://hal.archives-ouvertes.fr/lirmm-01370310

A. Gyárfás and J. Lehel, A Helly-type problem in trees, Combinatorial Theory and its applications, pp.571-584, 1969.

C. Archontia, . Giannopoulou, D. M. Pilipczuk, J. Thilikos, M. Raymond et al., Linear kernels for edge deletion problems to immersion-closed graph classes, 2016.

[. Grünwald, Ein Neuer Beweis Eines Mengerschen Satzes, Journal of the London Mathematical Society, vol.1, issue.3, pp.188-192, 1938.
DOI : 10.1112/jlms/s1-13.3.188

[. Guenin and R. Thomas, Packing directed circuits exactly, Combinatorica, vol.27, issue.4, pp.397-421, 2011.
DOI : 10.1287/moor.27.2.361.328
URL : http://arxiv.org/pdf/1012.2749

R. Halin, Tree-partitions of infinite graphs, Discrete Mathematics, vol.97, issue.1-3, pp.203-217, 1991.
DOI : 10.1016/0012-365X(91)90436-6
URL : https://doi.org/10.1016/0012-365x(91)90436-6

[. Haxell, Packing and covering triangles in graphs, Discrete Mathematics, vol.195, issue.1-3, pp.251-254, 1999.
DOI : 10.1016/S0012-365X(98)00183-6
URL : https://hal.archives-ouvertes.fr/lirmm-00806743

[. Huynh, F. Joos, and P. Wollan, A unified Erd? os?Pósa theorem for constrained cycles. ArXiv preprints, 2016.

P. Haxell and Y. Kohayakawa, Packing and Covering Triangles in Tripartite Graphs, Graphs and Combinatorics, vol.14, issue.1, pp.1-10, 1988.
DOI : 10.1007/s003730050010
URL : http://www.ime.usp.br/~yoshi/pronex/publications/MS/Journals/J5/paper.ps.gz

[. Haxell, A. Kostochka, and S. Thomassé, Packing and Covering Triangles in K 4-free Planar Graphs, Graphs and Combinatorics, vol.6, issue.5, pp.653-662, 2011.
DOI : 10.1007/BF01787705
URL : https://hal.archives-ouvertes.fr/lirmm-00806743

[. Havet and A. K. Maia, On disjoint directed cycles with prescribed minimum lengths, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00816135

[. Hochstättler, R. Nickel, and B. Peis, Two disjoint negative cycles in a signed graph, Electronic Notes in Discrete Mathematics, vol.25, pp.107-111, 2006.
DOI : 10.1016/j.endm.2006.06.084

A. Hajnal and J. Surányi, ¨ Uber die auflösung von graphen in vollständige teilgraphen, pp.113-121, 1958.

F. Joos, Parity linkage and the Erd? os-Pósa property of odd cycles through prescribed vertices in highly connected graphs. ArXiv preprint arXiv:1411, 2014.
DOI : 10.1002/jgt.22103

M. Bart, M. Jansen, and . Pilipczuk, Approximation and kernelization for chordal vertex deletion. arXiv preprint, to appear in the Proceedings of the 28th ACM-SIAM Symposium on Discrete Algorithms, 2016.

T. Johnson, N. Robertson, P. D. Seymour, and R. Thomas, Directed Tree-Width, Journal of Combinatorial Theory, Series B, vol.82, issue.1, pp.138-154, 2001.
DOI : 10.1006/jctb.2000.2031
URL : https://doi.org/10.1006/jctb.2000.2031

[. Kaiser, Transversals of d-Intervals, Discrete & Computational Geometry, vol.18, issue.2, pp.195-203, 1997.
DOI : 10.1007/PL00009315

[. Kakimura and K. Kawarabayashi, Packing Directed Circuits through Prescribed Vertices Bounded Fractionally, SIAM Journal on Discrete Mathematics, vol.26, issue.3, pp.1121-1133, 2012.
DOI : 10.1137/100786423

[. Kakimura and K. Kawarabayashi, Half-integral packing of odd cycles through prescribed vertices, Combinatorica, vol.31, issue.5, pp.549-572, 2013.
DOI : 10.1007/s00493-011-2551-5

[. Kakimura and K. Kawarabayashi, Fixed-parameter tractability for subset feedback set problems with parity constraints, Theoretical Computer Science, vol.576, pp.61-76, 2015.
DOI : 10.1016/j.tcs.2015.02.004

[. Kawarabayashi and Y. Kobayashi, Edge-disjoint odd cycles in 4-edge-connected graphs, Journal of Combinatorial Theory, Series B, vol.119, pp.12-27, 2016.
DOI : 10.1016/j.jctb.2015.12.002
URL : https://hal.archives-ouvertes.fr/hal-00678188

K. Kawarabayashi, D. Král, M. Kr?ál, and S. Kreutzer, Packing directed cycles through a specified vertex set, Proceedings of the Twenty- Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, pp.365-377, 2013.
DOI : 10.1137/1.9781611973105.27
URL : https://epubs.siam.org/doi/pdf/10.1137/1.9781611973105.27

[. Kakimura, K. Kawarabayashi, and D. Marx, Packing cycles through prescribed vertices, Journal of Combinatorial Theory, Series B, vol.101, issue.5, pp.378-381, 2011.
DOI : 10.1016/j.jctb.2011.03.004
URL : https://doi.org/10.1016/j.jctb.2011.03.004

[. Kloks, C. M. Lee, and J. Liu, New Algorithms for k-Face Cover, k-Feedback Vertex Set, and k-Disjoint Cycles on Plane and Planar Graphs, 28th International Workshop on Graph Theoretic Concepts in Computer Science volume 2573 of LNCS, pp.282-295, 2002.
DOI : 10.1007/3-540-36379-3_25

, Tom Kloks. Treewidth. Computations and Approximations LNCS, vol.842, 1994.

O. Kwon and D. Marx, Erd? os?Pósa property of planar-H-minor models with prescribed vertex sets, 2015.

[. Kawarabayashi and A. Nakamoto, The Erd??s???P??sa property for vertex- and edge-disjoint odd cycles in graphs on orientable surfaces, Discrete Mathematics, vol.307, issue.6, pp.764-768, 2007.
DOI : 10.1016/j.disc.2006.07.008

[. Kawarabayashi and B. Reed, Highly parity linked graphs, Combinatorica, vol.29, issue.2, pp.215-225, 2009.

M. Krivelevich, On a conjecture of Tuza about packing and covering of triangles, Discrete Mathematics, vol.142, issue.1-3, pp.281-286, 1995.
DOI : 10.1016/0012-365X(93)00228-W

D. Král, '. , and H. Voss, Edge-disjoint odd cycles in planar graphs, Journal of Combinatorial Theory, Series B, vol.90, issue.1, pp.107-120, 2004.
DOI : 10.1016/S0095-8956(03)00078-9

[. Kawarabayashi and P. Wollan, Non-zero disjoint cycles in highly connected group labelled graphs, Journal of Combinatorial Theory, Series B, vol.96, issue.2, pp.296-301, 2006.
DOI : 10.1016/j.jctb.2005.08.001

[. Liu, Packing and Covering Immersions in 4-Edge-Connected Graphs. ArXiv preprint, 2015.

L. Lovász, On two minimax theorems in graph, Journal of Combinatorial Theory, Series B, vol.21, issue.2, pp.96-103, 1976.
DOI : 10.1016/0095-8956(76)90049-6

[. Liu, L. Postle, and P. Wollan, Erd? os-Pósa property for the topological minors, 2014.

C. L. Lucchesi and D. H. Younger, A Minimax Theorem for Directed Graphs, Journal of the London Mathematical Society, vol.2, issue.3, pp.369-374, 1978.
DOI : 10.1112/jlms/s2-17.3.369

[. Mader, ???ber die Maximalzahl kantendisjunkterA- Wege, Archiv der Mathematik, vol.28, issue.1, pp.325-336, 1978.
DOI : 10.1007/BF01226062

[. Mader, ???ber die Maximalzahl kreuzungsfreierH-Wege, Archiv der Mathematik, vol.30, issue.1, pp.387-402, 1978.
DOI : 10.1007/BF01226465

K. Menger, Zur allgemeinen Kurventheorie, Fundamenta Mathematicae, vol.10, issue.10, pp.96-115, 1927.
DOI : 10.4064/fm-10-1-96-115

[. Mousset, A. Noever, F. Nemanja?kori´cnemanja?nemanja?kori´nemanja?kori´c, and . Weissenberger, A tight Erd? os-Pósa function for long cycles, 2016.

[. Mohar and C. Thomassen, Graphs on surfaces, 2001.

D. Marx and P. Wollan, An exact characterization of tractable demand patterns for maximum disjoint path problems, Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp.642-661, 2015.
DOI : 10.1137/1.9781611973730.44

J. Ma, X. Yu, and W. Zang, Approximate min-max relations on plane graphs, Journal of Combinatorial Optimization, vol.16, issue.1, pp.127-134, 2013.
DOI : 10.1007/BF01271272

. Nl-]-víctor-neumann-lara, Cyclic transversality for even and long cycles

M. Pontecorvi and P. Wollan, Disjoint cycles intersecting a set of vertices, Journal of Combinatorial Theory, Series B, vol.102, issue.5, pp.1134-1141, 2012.
DOI : 10.1016/j.jctb.2012.05.004
URL : http://wwwusers.di.uniroma1.it/%7Ewollan/PUBS/sigma_cyc_web.pdf

A. Bruce and . Reed, Tree width and tangles: A new connectivity measure and some applications, pp.87-162, 1997.

A. Bruce and . Reed, Mangoes and blueberries, Combinatorica, vol.19, issue.2, pp.267-296, 1999.

D. Rautenbach and B. Reed, The Erd??s-P??sa Property for Odd Cycles in Highly Connected Graphs, Combinatorica, vol.21, issue.2, pp.267-278, 2001.
DOI : 10.1007/s004930100024

A. Bruce, N. Reed, P. D. Robertson, R. Seymour, and . Thomas, Packing directed circuits, Combinatorica, vol.16, issue.4, pp.535-554, 1996.

[. Robertson and P. D. Seymour, Graph minors. V. Excluding a planar graph, Journal of Combinatorial Theory, Series B, vol.41, issue.1, pp.92-114, 1986.
DOI : 10.1016/0095-8956(86)90030-4
URL : https://doi.org/10.1006/jctb.1999.1919

A. Bruce, B. F. Reed, and . Shepherd, The Gallai-Younger conjecture for planar graphs, Combinatorica, vol.16, issue.4, pp.555-566, 1996.

[. Robertson, P. Seymour, and R. Thomas, Quickly Excluding a Planar Graph, Journal of Combinatorial Theory, Series B, vol.62, issue.2, pp.323-348, 1994.
DOI : 10.1006/jctb.1994.1073
URL : https://doi.org/10.1006/jctb.1994.1073

. Jean-florent, . Raymond, . Ignasi, D. M. Sau, and . Thilikos, An edge variant of the Erd? os-Pósa property, Discrete Mathematics, vol.339, issue.8, pp.2027-2035, 2016.

J. Raymond and D. M. Thilikos, Polynomial Gap Extensions of the Erd? os-Pósa Theorem ArXiv preprint arXiv:1305.7376, full version of, 2013.

J. Raymond and D. M. Thilikos, Polynomial gap extensions of the Erd? os-Pósa theorem, The Seventh European Conference on Combinatorics, Graph Theory and Applications, pp.13-18, 2013.
DOI : 10.1007/978-88-7642-475-5_3

A. Schrijver, A Short Proof of Mader's S-Paths Theorem, Journal of Combinatorial Theory, Series B, vol.82, issue.2, pp.319-321, 2001.
DOI : 10.1006/jctb.2000.2029
URL : https://doi.org/10.1006/jctb.2000.2029

D. Seese, Tree-partite graphs and the complexity of algorithms, Proceedings of Fundamentals of Computation Theory, pp.412-421, 1985.
DOI : 10.1007/BFb0028825

P. D. Seymour, Packing circuits in eulerian digraphs, Combinatorica, vol.64, issue.2, pp.223-231, 1996.
DOI : 10.1007/BF01844848

[. Simonovits, A new proof and generalizations of a theorem of Erd??s and P??sa on graphs withoutk+1 independent circuits, Acta Mathematica Academiae Scientiarum Hungaricae, vol.9, issue.1-2, pp.191-206, 1967.
DOI : 10.1007/BF02020974

A. Seb?-o and L. Szeg?-o, The path-packing structure of graphs, Integer Programming and Combinatorial Optimization: 10th International IPCO Conference, pp.256-270, 2004.

C. Thomassen, Girth in graphs, Journal of Combinatorial Theory, Series B, vol.35, issue.2, pp.129-141, 1983.
DOI : 10.1016/0095-8956(83)90067-9

C. Thomassen, On the presence of disjoint subgraphs of a specified type, Journal of Graph Theory, vol.10, issue.1, pp.101-111, 1988.
DOI : 10.4153/CJM-1965-035-8

C. Thomassen, The Erd??s-P??sa Property for Odd Cycles in Graphs of Large Connectivity, Combinatorica, vol.21, issue.2, pp.321-333, 2001.
DOI : 10.1007/s004930100028

Z. Tuza, A conjecture on triangles of graphs, Graphs and Combinatorics, vol.24, issue.4, pp.373-380, 1990.
DOI : 10.1007/BF01787705

[. Voss, Some properties of graphs containing k independent circuits, Proceedings of Colloquium Tihany, pp.321-334, 1968.

P. Wollan, Packing cycles with modularity constraints, Combinatorica, vol.100, issue.2, pp.95-126, 2011.
DOI : 10.1016/j.jctb.2009.05.003

P. Wollan, The structure of graphs not admitting a fixed immersion, Journal of Combinatorial Theory, Series B, vol.110, pp.47-66, 2015.
DOI : 10.1016/j.jctb.2014.07.003